Quantum Gravity

8QUANTUM COSMOLOGY8.1 Minisuperspace models8.1.1 General introductionAs we have discussed at length in Chapters 5 and 6, all information about canonicalquantum gravity lies in the constraints (apart from possible surface terms).These constraints assume in the Dirac approach the form of conditions for physicallyallowed wave functionals. In quantum geometrodynamics (Chapter 5), thewave functional depends on the three-metric, while in quantum connection orquantum loop dynamics (Chapter 6), it depends on a non-Abelian connection ora Wilson-loop-type variable.A common feature for all variables is that the quantum constraints are difficultif not impossible to solve. In classical GR, the field equations often becometractable if symmetry reductions are performed; one can, for example, imposespherical symmetry, axial symmetry, or homogeneity. This often corresponds tointeresting physical situations: stationary black holes are spherically or axiallysymmetric (Section 7.1), while the Universe as a whole can be approximated byhomogeneous and isotropic models. The idea is to apply a similar procedure inthe quantum theory. One may wish to make a symmetry reduction at the classicallevel and to quantize only a restricted set of variables. The quantization ofblack holes discussed in Chapter 7 is a prominent example. The problem is thatsuch a reduction violates the uncertainty principle, since degrees of freedom areneglected together with the corresponding momenta. Still, the reduction may bean adequate approximation in many circumstances. In quantum mechanics, forexample, the model of a ‘rigid top’ is a good approximation as long as otherdegrees of freedom remain unexcited due to energy gaps. Such a situation canalso hold for quantum gravity; cf. Kuchař and Ryan (1989) for a discussion ofthis situation in a quantum-cosmological context. In the dynamical-triangulationapproach discussed in Section 2.2.6, for example, one can derive an effective cosmologicalaction from the full path integral (Ambjørn et al. 2005).Independent of this question whether the resulting models are realistic or not,additional reasons further support the study of dimensionally reduced models.First, they can play the role of toy models to study conceptual issues which areindependent of the number of variables. Examples are the problem of time, therole of observers, and the emergence of a classical world; cf. Chapter 10. Second,they can give the means to study mathematical questions such as the structureof the wave equation and the implementation of boundary conditions. Third, onecan compare various quantization schemes in the context of simple models. This243